#8107609

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#43982	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(2,5)(3,9)(8,12)(10,15)(17,18,23,25)(19,28,30,24)(20,31,26,21)(22,32,29,27)(33,34,39,44)(35,40,46,47)(36,43,38,41)(37,42,48,45);(1,2,6,5)(3,7,9,11)(4,8,13,12)(10,14,15,16)(17,19,23,30)(18,24,25,28)(20,32,26,27)(21,29,31,22)(33,35,39,46)(34,40,44,47)(36,42,38,45)(37,43,48,41);(3,9)(7,11)(10,15)(14,16)(18,25)(20,26)(24,28)(27,32)(33,36,39,38)(34,41,44,43)(35,45,46,42)(37,40,48,47);(1,3,6,9)(2,7,5,11)(4,10,13,15)(8,14,12,16)(17,20,23,26)(18,21,25,31)(19,27,30,32)(22,28,29,24)(33,37,39,48)(34,42,44,45)(35,41,46,43)(36,47,38,40);(1,4,6,13)(2,8,5,12)(3,10,9,15)(7,14,11,16)(17,21)(18,26)(19,29)(20,25)(22,30)(23,31)(24,27)(28,32)(33,38,39,36)(34,43,44,41)(35,45,46,42)(37,40,48,47);(1,5,6,2)(3,11,9,7)(4,12,13,8)(10,16,15,14)(17,22,23,29)(18,27,25,32)(19,21,30,31)(20,28,26,24)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014805);